Thierry Gallay
Interaction of vortices in viscous planar flows
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ABSTRACT. We consider the inviscid limit for the two-dimensional
incompressible Navier-Stokes equation in the particular case where
the initial flow is a finite collection of point vortices. We
suppose that the initial positions and the circulations of the
vortices do not depend on the viscosity parameter $\nu$, and we
choose a time $T > 0$ such that the Helmholtz-Kirchhoff point vortex
system is well-posed on the interval $[0,T]$. Under these
assumptions, we prove that the solution of the Navier-Stokes
equation converges, as $\nu \to 0$, to a superposition of Lamb-Oseen
vortices whose centers evolve according to a viscous regularization
of the point vortex system. Convergence holds uniformly in time, in
a strong topology which allows to give an accurate description of
the asymptotic profile of each individual vortex. In particular, we
compute to leading order the deformations of the vortices due to
mutual interactions. This allows to estimate the self-interactions,
which play an important role in the convergence proof.